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In this paper, we consider a general time-inconsistent optimal controlproblem for a non homogeneous linear system, in which its state evolvesaccording to a stochastic differential equation with deterministiccoefficients, when the noise is driven by a Brownian motion and an independentPoisson point process. The running and the terminal costs in the objectivefunctional, are explicitly dependent on some general discounting coefficientswhich cover the non-exponential and the hyperbolic discounting situations.Furthermore, the presence of some quadratic terms of the conditionalexpectation of the state process as well as a state-dependent term in theobjective functional makes the problem time-inconsistent. Open-loop Nashequilibrium controls are constructed instead of optimal controls, by using aversion of the stochastic maximum principle approach. This approach involves astochastic system that consists of a flow of forward-backward stochasticdifferential equations and an equilibrium condition. As an application, westudy some concrete examples.

The Maximum Principle in Time-Inconsistent LQ Optimal Control Problem for Jump Diffusions